### June 2006: Shaking hands (solution)

This is the solution to the Shaking Hands puzzle.

Mr. Dickinson shook hands with 4 people.

Suppose the other four couples are Mr. and Mrs. A, Mr. and Mrs. B, Mr. and Mrs. C, and Mr. and Mrs. D. We know Mr. Dickinson did not shake hands with 8 people, because if he did, then everyone else would have shaken hands with at least one person, and we know that someone shook zero hands. Say that it was Mr. A who shook 8 hands. Then Mrs. A must have shaken 0 (in the graph below, lines indicate hand shakes). By a similar logic, Mr. Dickinson couldn't have shaken 7 hands, because then no one could have shaken 1 hand. Say Mr. B shook 7 hands. Then Mrs. B shook 1 hand. Arguing in the same way, Mr. Dickinson couldn't have shaken 6 hands, so someone else did---Mr. C, say. Then Mrs. C shook 2 hands. And again, someone else shook 5 hands, Mr. D, and his wife, Mrs. D, shook 3 hands. Thus, Mr. Dickinson shook 4 hands.

This month's winners: Evan Templeton, Ryan Deeds, Jared Lease, Shashwat Acharya, Hiro Arai

Mr. Dickinson shook hands with 4 people.

Suppose the other four couples are Mr. and Mrs. A, Mr. and Mrs. B, Mr. and Mrs. C, and Mr. and Mrs. D. We know Mr. Dickinson did not shake hands with 8 people, because if he did, then everyone else would have shaken hands with at least one person, and we know that someone shook zero hands. Say that it was Mr. A who shook 8 hands. Then Mrs. A must have shaken 0 (in the graph below, lines indicate hand shakes). By a similar logic, Mr. Dickinson couldn't have shaken 7 hands, because then no one could have shaken 1 hand. Say Mr. B shook 7 hands. Then Mrs. B shook 1 hand. Arguing in the same way, Mr. Dickinson couldn't have shaken 6 hands, so someone else did---Mr. C, say. Then Mrs. C shook 2 hands. And again, someone else shook 5 hands, Mr. D, and his wife, Mrs. D, shook 3 hands. Thus, Mr. Dickinson shook 4 hands.

This month's winners: Evan Templeton, Ryan Deeds, Jared Lease, Shashwat Acharya, Hiro Arai